Real nullstellensatz and *-ideals in *-algebras

Abstract

For a fixed tuple of square matrices X ={X_1,...,X_g} the set I(X) of all noncommutative polynomials p in X and X∗ such that p(X) = 0 is an ideal in the ∗-algebra of all polynomials. This article concerns such zeroes and their corresponding ideals. An algebraic characterization of ideals of the form I(X) is a real nullstellensatz. A main result of this article is a strong nullstellensatz for a ∗-ideal of finite codimension in a ∗-algebra. Without the finite codimension assumption, there are examples of such ideals which do not satisfy, very liberally interpreted, any Nullstellensatz. A polynomial p in noncommuting variables (x_1,...,x_g,x∗_1,...,x_∗g) is called analytic if it is a polynomial in the variables x_j only. As shown in this article, ∗-ideals generated by analytic polyno-mials do satisfy a natural Nullstellensatz and those generated by homogeneous analytic polynomials have a particularly simple description. Another natural notion of zero of a noncommutative polynomial p is a pair (X, v) such that p(X)v = 0; here X is an n by n matrix tuple and v ∈ R^n. For fixed (X,v), the set of all such polynomials is a left ideal. The relationship between such zeroes and their left ideals is considerably more developed than is our beginning effort here. This article provides a guide to that literature.